Hans Hahn (mathematician)
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Hans Hahn (mathematician)
Hans Hahn (; 27 September 1879 – 24 July 1934) was an Austrian mathematician and philosopher who made contributions to functional analysis, topology, set theory, the calculus of variations, real analysis, and order theory. In philosophy he was among the main logical positivists of the Vienna Circle. Biography Born in Vienna as the son of a higher government official of the k.k. Telegraphen-Korrespondenz Bureau (since 1946 named "Austria Presse Agentur"), in 1898 Hahn became a student at the Universität Wien starting with a study of law. In 1899 he switched over to mathematics and spent some time at the universities of Strasbourg, Munich and Göttingen. In 1902 he took his Ph.D. in Vienna, on the subject "Zur Theorie der zweiten Variation einfacher Integrale". He was a student of Gustav von Escherich. He was appointed to the teaching staff (Habilitation) in Vienna in 1905. After 1905/1906 as a stand-in for Otto Stolz at Innsbruck and some further years as a Privatdozent in Vie ...
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Hahn Embedding Theorem
In mathematics, especially in the area of abstract algebra dealing with ordered structures on abelian groups, the Hahn embedding theorem gives a simple description of all linearly ordered abelian groups. It is named after Hans Hahn. Overview The theorem states that every linearly ordered abelian group ''G'' can be embedded as an ordered subgroup of the additive group ℝΩ endowed with a lexicographical order, where ℝ is the additive group of real numbers (with its standard order), Ω is the set of ''Archimedean equivalence classes'' of ''G'', and ℝΩ is the set of all functions from Ω to ℝ which vanish outside a well-ordered set. Let 0 denote the identity element of ''G''. For any nonzero element ''g'' of ''G'', exactly one of the elements ''g'' or −''g'' is greater than 0; denote this element by , ''g'', . Two nonzero elements ''g'' and ''h'' of ''G'' are ''Archimedean equivalent'' if there exist natural numbers ''N'' and ''M'' such that ''N'', ''g'', &nb ...
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Calculus Of Variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as '' geodesics''. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depend ...
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